The trails that currently exist  in the park are paved.  More paths are under construction.  Also, just north of the park is Storza Woods, a 15 acre hardwood forest.  The trail here is probably less than a mile loop, but well maintained.  Storza Woods is located just inside the entrance to the Atlanta Botanical Gardens.

The Atlanta Track Club sponsors a couple of excellent cross-country races in the park.  These races typically take place in October and December.  So, running in the park off the pavement can be done.

Directions:  From Atlanta, take I-75/I-85 to 10th Street. Turn right onto 10th street (go east).  Go less than a mile, the park will be on your left.  The Botanical Gardens are located on Piedmont Ave. just north of the park.  The gate to the trail is open from 9AM to 6PM (till 7PM from April-October)

Parking:  You are on your own.  Only street parking is available.  Please be considerate of the homeowners in this neighborhood, do not block their driveways.
Atlanta Botanical Gardens web site